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aliases: ["periodic orbits"]

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# Motivation

*Open Problem:**
Does every triangular billiards admit a [periodic orbit.md](periodic orbit.md)?

*Answer (1775):* 
Yes for acute triangles, there is at least one periodic orbit:

![](attachments/2020-02-01-23-59-05.png)

For arbitrary triangles: unknown!

Let $M$ be a [Hamiltonian.md](Hamiltonian.md).

For [regular value) $r\in \RR$ of the  Hamiltonian, $H\inv(r](regular value) $r/in /RR$ of the  Hamiltonian, $H/inv(r) \subset M$ is a submanifold $Y\subset M$ with a smooth vector field $X_H$ called a *regular level set*.

**Question:**
Does $X_H$ have a closed orbit on every regular level set?
What conditions do you need to guarantee the existence of a closed orbit?

Turns out not to depend on $H$, and only on the hypersurface $Y$.
The existence of a closed orbit is equivalent to the existence of a closed embedded curve $\gamma$ that is everywhere tangent to $\ker(\restrictionof{\omega}{Y})$.

**Question:**
When is such a curve guaranteed to exist?

**Theorem (Weinstein, 1972):** 
If $Y$ is convex.

**Theorem (Rabinowitz)**: 
If $Y$ is "star-shaped" (exists a point $p$ that can "see" all points via straight lines).

**Theorem (1987):**
Every contact-type hypersurface in the [periodic orbit.md](periodic orbit.md).

**Conjecture (Weinstein, 1978):**
Let $(M, \xi)$ be a closed (compact) [periodic orbit.md](periodic orbit.md).

**Theorem (Weinstein, Dimension 3, Overtwisted. 1993):**
Let $(M, \lambda, \xi)$ be a closed [overtwisted contact structure.md](overtwisted contact structure.md). 
Then the [periodic orbit.md](periodic orbit.md).